let-u-i-j-k-and-v-2i-j-3k-o-i-j-k-orthonormal-1-calculate-u-v-u-v-2-calculate-cos-u-v-3-calculate-u-v-and-sin-u-v-

Question Number 86406 by mathmax by abdo last updated on 28/Mar/20

l e t \vec{u} = \vec{i} - \vec{j} + \vec{k} a n d \vec{v} = 2 \vec{i} + \vec{j} + 3 \vec{k}

(o, i, j, k) o r t h o n o r m a l

1) c a l c u l a t e ∣∣ \vec{u} ∣∣, ∣∣ \vec{v} ∣∣, \vec{u} . \vec{v}

2) c a l c u l a t e c o s (\vec{u}, \vec{v})

3) c a l c u l a t e \vec{u} Λ \vec{v} a n d s i n (\vec{u}, \vec{v})

Answered by jagoll last updated on 28/Mar/20

∣∣ \vec{u} ∣∣ = \sqrt{3}

∣∣ \vec{v} ∣∣ = \sqrt{14}

\vec{u} . \vec{v} = 2 - 1 + 3 = 4

\cos (\vec{u}, \vec{v}) = \frac{4}{\sqrt{42}}

Answered by Rio Michael last updated on 28/Mar/20

3) ∣ \vec{u} \times \vec{v} ∣ = ∣ \vec{u} ∣ ∣ \vec{v} ∣ \sin θ

\Rightarrow \sin θ = \frac{∣ \vec{u} \times \vec{v} ∣}{∣ \vec{u} ∣∣ \vec{v} ∣}

\vec{u} \times \vec{v} = | \begin{matrix} \vec{i} & \vec{j} & \vec{k} \\ 1 & - 1 & 1 \\ 2 & 1 & 3 \end{matrix} | = i | \begin{matrix} - 1 & 1 \\ 1 & 3 \end{matrix} | - j | \begin{matrix} 1 & 1 \\ 2 & 3 \end{matrix} | + k | \begin{matrix} 1 & - 1 \\ 2 & 1 \end{matrix} |

= - 4 i - j + 3 k

∣ \vec{u} \times \vec{v} ∣ = \sqrt{26}

∣ \vec{u} ∣ = \sqrt{3} and ∣ \vec{v} ∣ = \sqrt{14} \Rightarrow \sin (\vec{u}, \vec{v}) = \frac{\sqrt{26}}{\sqrt{3} \sqrt{14}} = \sqrt{\frac{13}{14}}

Commented by abdomathmax last updated on 28/Mar/20

t h a n k x s i r .